A common way to summarize data is to use numerical summaries, many of which are based on the ordered data. For example, the largest and smallest data values (minimum and maximum) are the first and last values in the ordered data. If we know the first and last values in an ordered list, we know that all the data values are between these two numbers.

Another numerical summary that is based on ordered data is the median, which is the middle value in an ordered list. Let's find and interpret the median, using our raisin data. Note 8

Which position corresponds to the median (the position in the middle of the list)? How many raisins are there in the box at this position?

One way to find the median is to continue to remove the highest and lowest values in the data set until only the median remains. Close Tip

One way to find the median is to continue to remove the highest and lowest values in the data set until only the median remains.

The median is the value in the exact center of a data set -- in other words there are as many values above it as there are below it. In this case, the median is in the ninth position, since there are eight values below it and eight above. Note that in any data set with 17 values, the ninth value in the ordered list will always be the median.

Problem D2

Suppose that the data were ordered from highest to lowest, instead of from lowest to highest. How would you find the median then?

We can use the median along with the minimum and maximum to describe variation in data. The median divides the raisin-count data into two groups: the data values below the median and the data values above the median. Note that each group has eight data values, which is approximately half the data. Consequently, approximately half of the raisin-counts are in the interval 25 to 28 (from the minimum to the median), and approximately half of the raisin counts are in the interval 28 to 31 (from the median to the maximum).

We can also determine the median by looking at a line plot. For the line plot of the raisin counts, you can identify the 17 positions in the ordered data as follows:

Alternatively, you could number the 17 positions in this way:

These two line plots are identical, since there is no need to distinguish the order of raisin boxes that have the same number of raisins.

Again we see that the ordered lists contain 17 data values, and there are 17 positions in the ordered list. Position 9 contains the middle value because eight positions precede Position 9, and eight positions follow Position 9. Position 9 corresponds to a box that contains 28 raisins, so 28 is the median. This can be determined from either of the line plots above.